G-module

The torus can be made an abelian group isomorphic to the product of the circle group. This abelian group is a Klein four-group-module, where the group acts by reflection in each of the coordinate directions (here depicted by red and blue arrows intersecting at the identity element).

The collection of left (respectively right) G-modules and their morphisms form an abelian categoryG-Mod (resp. Mod-G). The category G-Mod (resp. Mod-G) can be identified with the category of left (resp. right) ZG-modules, i.e. with the modules over the group ringZ[G].

A submodule of a G-module M is a subgroup A ⊆ M that is stable under the action of G, i.e. g·a ∈ A for all g ∈ G and a ∈ A. Given a submodule A of M, the quotient moduleM/A is the quotient group with action g·(m + A) = g·m + A.